The measurement of the properties of a fluid, such as its surface tension in a surrounding liquid or gas, is needed in many applications and across many industries including the food, textile, chemical, oil, pharmaceutical, biological and electronic industries, and in research institutions working in these fields. Similar to the surface tension measurement, other geometric properties of a bubble or drop can be measured using the same methodology, such as the bubble/drop volume, interface area, apex curvature radius, or height of centre of gravity. The angle at the base of the bubble/drop (known as contact angle) can also be measured similarly. This latter measurement is particularly useful in measuring the wettability property between a solid surface and a fluid; something that is vital in many engineering applications.
A common method of estimating surface tension involves accurately measuring and analysing the shape of a drop (or bubble) of the fluid. This shape is defined mathematically by a complex differential equation that involves the surface tension property as well as the respective fluid densities and gravity, which can be measured in other ways or looked up from a table of known values. By regression fitting a numerically integrated mathematical solution of the equation to the experimentally measured shape of the drop, the value of the surface tension is calculated.
FIG. 1 shows an experimental set-up for measuring the surface tension of an unknown transparent liquid 10. A gas bubble 12 is formed from an upward-facing horizontal orifice of a needle 14 of a syringe 16 into the unknown liquid 10. The bubble 12 is slowly and steadily injected, or simply steadily attached to the rim of the orifice, so that its shape corresponds to a static bubble shape. A camera 18 is used to capture the image of the bubble shape, illuminated by a backlight 20.
FIG. 2 shows an image captured from such a system. Image processing software can extract from this image a curve which approximates the profile of the bubble-liquid interface, subject to the constraints of the camera resolution and difference between the actual profile when viewed in a true mathematical elevation versus that captured by a camera capturing the bubble from a viewing angle that cannot quite see both opposite edges of the profile
From a physical point of view, the shape of such an axisymmetric bubble is a result of the hydrostatic pressure gradients in both the liquid and gas phases and of the capillary equilibrium at the liquid-gas interface. The mathematical equation of the bubble profile can be written as follows (see F. J. Lesage, J. S. Cotton and A. J. Robinson, Analysis of quasi-static vapour bubble shape during growth and departure, Physics of Fluids, Vol. 25, p. 067103, 2013):
                                                                        (                                                      ρ                    l                                    -                                      ρ                    g                                                  )                            ⁢              g                        σ                    ⁢          z                =                              2                          R              0                                -                      C            ⁡                          (              z              )                                                          Eq        .                                  ⁢        1            
In equation 1, ρl, ρg and a are properties of the fluid, respectively the liquid and gas densities and the interfacial surface tension; g is the gravitational acceleration; R0 is the radius of curvature at the apex of the bubble; z is the vertical coordinate, downward from the apex of the bubble, and C is the curvature of the interface, which depends on the vertical coordinate (thus expressed as C(z)). The profile of the bubble is fully defined by this equation, and is cut by a horizontal plane which is the horizontal surface on which the bubble is attached.
More generally, the terms ρl, ρg can be replaced by the densities of any two fluids and are not necessarily those of a liquid and a gas. Equation 1 can therefore be understood as covering the more general case of two fluids, not necessarily a liquid and a gas.
The interfacial surface tension, σ, is a property of the two fluids at the interface and therefore will be different for e.g. a water droplet in a heavy oil medium than it will for water in a gaseous medium. While interfacial surface tension is strictly speaking thus a property of the liquid and the gas, all gases effectively behave the same way, and so in the specific case of a liquid/gas interaction the surface tension is generally considered as a property specific to the liquid.
This equation is based on the assumption that the bubble or droplet is not freely floating but rather is attached to a surface. (Note that droplets and bubbles are, for these purposes, different examples of the same phenomenon, namely a discrete body of a first fluid disposed in a second fluid.) The droplet/bubble can be gravitationally accelerated, due to weight and buoyancy forces arising from the density difference with the surrounding fluid, either towards the surface in question, in which case it is a “sessile” droplet or bubble; or away from the surface, in which case it is a “pendant” droplet or bubble, remaining attached by surface tension forces that oppose the gravitationally-induced buoyancy or weight.
The bubble shown in FIG. 1 is pendant, and is equivalent to the classic image of a droplet of water hanging from a tap (or faucet) prior to it breaking off when it grows beyond the size limit allowing it to remain attached under surface tension. An example of a sessile droplet is a bead of water lying on a polished surface, while sessile bubbles would include bubbles of air trapped under a submerged glass surface. Note that while these simple examples all presume a liquid-gas or gas-liquid system of two fluids, the concepts apply equally to any two immiscible fluids of different densities. The bodies of a first fluid disposed in a second fluid will be referred to herein as droplets/bubbles, droplets, or bubbles, depending on context, it being appreciated that the terms can be used interchangeably from the point of view of the physics involved.
Equation 1 therefore provides a relation that links different properties of the two fluids and the gravitational acceleration to the geometry of the droplet/bubble. Thus, if sufficient independent parameters are known or measured, the other parameters can be deduced using this equation.
The usual way of calculating the surface tension of a fluid is as follows. The gravitational acceleration and the fluid densities are considered as known (or measured with a different method). Then, different geometrical profiles corresponding to equation 1 are generated by numerical integration of equation 1. Each numerical integration provides a solution which describes the entire surface of the bubble; from the base to the tip. A regression algorithm is utilized to obtain the solution that best fits the experimental profile, which is obtained by image processing. The best fitting profile provides the value of the surface tension property.
Similarly, the volume of the drop or bubble, or the height/position of its centre of gravity can be calculated when a best fit geometric profile is obtained describing the curvature of the profile.
Such methods have the drawback that they require significant image processing capability and mathematical computational power, as well as a good deal of post-processing time to obtain a good fit between the mathematical model and the experimental image.